Questions tagged [elliptic-functions]
The elliptic-functions tag has no summary.
102 questions
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How to construct elliptic functions with predescribled zeros and poles by means of Weierstrass ℘-function and its derivatives?
Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
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195
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Rigorous analysis of an elliptic integration arising from physics
Let $\delta_+,\delta_-$ be two complex variables. Denote $e_i, i = 1,...,6$ be (multi-valued) holomorphic functions of $\delta_+,\delta_-$ defined as follows:
$$
\begin{align}
e_1 & = -2\left(1 - \...
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Are $j(\tau)$ and $\wp \left(\frac{\omega_1}{n},\omega_1,\omega_2\right)$ always expressible by radicals over $\mathbb{Q}$?
Let $j$ be the j-invariant defined by
$$j(\tau)=\frac{E_4(\tau)^3}{\Delta(\tau)},\quad \Im(\tau)>0$$
where
$$E_4(\tau)=1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n},\quad q=e^{2\pi i\tau},$$
$$\...
- 331
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129
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How to solve any lemnatomic equation?
A cyclotomic equation is the equation obtained by setting the minimal polynomial, over $\mathbb{Q}$, of $\mathrm{e}^{2\pi i/n}$ equal to zero.
Gauss wrote about solving cyclotomic equations by ...
- 331
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2
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2k
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What is this modified arithmetico-geometric mean function?
I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
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152
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The cubic root of modular functions
Assume $p\neq 3$ is a prime. Let $f$ be a modular function on $X_0(3p)$ whose divisor is support on the upper half plane (i.e. $f$ has no zeros and poles at the cusps). If the divisor $D(f)=3D$ where $...
- 538
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415
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Conflicting notation for periods of elliptic functions
It seems that there is absolutely no general agreement on the notation
for the periods $\omega_1$ and $\omega_2$ of Weierstrass elliptic functions.
Even if Serge Lang's book on elliptic functions is ...
- 14.1k
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2
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385
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Partial differential equations for the Weierstrass sigma function
The Wolfram Functions Site
gives for the Weierstrass sigma function two partial differential equation w.r.t. $g_2,g_3$. The first one is
$$ z\frac{\partial \sigma(z; g_2, g_3)}{\partial z}
- 4g_2\...
- 13.5k
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Solving the quintic using the eta quotients $\frac{\eta(\tau)}{\eta(2\tau)},\,\frac{\eta(\tau)}{\eta(3\tau)},\,\frac{\eta(\tau)}{\eta(4\tau)},$ etc?
I. Reduced quintics
The general quintic can be reduced to the one-parameter forms,
$$x^5+5x+\alpha=0\\[5pt]
x^5+5\alpha x^2-\alpha=0$$
for some generic alpha. The first is the Bring form and there are ...
- 13.1k
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answer
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Factorisation of division polynomial
Let $\Psi_n$ denote the $n$-th division polynomial associated with the elliptic curve $y^2 = x^3 + A$, where $n$ is a natural number. The division polynomials are defined recursively, as described in ...
- 359
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Using Ramanujan's cubic continued fraction $C(q)$ and $x^3+y^3=1$ to solve the Bring quintic?
The octic Ramanujan-Selberg continued fraction $S(q)$ and $x^8+y^8=1$ can solve the Bring quintic. So can the Rogers-Ramanujan continued fraction $R(q)$ and $x^5+y^5=1.\,$ It turns out Ramanujan's ...
- 13.1k
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Does Hermite's approach to the Bring quintic yield pairs of methods?
In a previous post, we mentioned $4+2=6$ methods to solve the Bring quintic. The first four uses the same quartic to find the elliptic modulus $k$ and the last two uses the same octic. For balance, we ...
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Solving the Bring quintic using the Ramanujan $g$- and $G$-functions?
Ramanujan defined two functions now called the Ramanujan g- and G-functions. One of the more well-known values is,
$$g_{58} = \sqrt{\tfrac{5+\sqrt{29}}2}$$
If we let,
$$2^6\big(g_{58}^{12}+g_{58}^{-12}...
- 13.1k
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Two new formulas to solve the Bring quintic using only Jacobi $\vartheta_3(q)$ and $\vartheta_4(q)$?
(Note: Emil Jann Fiedler found the formula for the Bring quintic using $R(q)$ in 2021, and these two formulas using $\vartheta_3(q)$ and $\vartheta_4(q)$ in 2022.) Recall the Jacobi theta functions,
$$...
- 13.1k
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Who solved the Bring quintic using the Rogers-Ramanujan continued fraction $R(q)$ and how to find all five roots?
I. The octahedral group
Given the nome $q=e^{\pi i \tau}$, then the elliptic lambda function $\lambda(\tau)$ shown below with its Ramanujan–Selberg continued fraction,
\begin{align}
\big(\lambda(\tau)...
- 13.1k
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Is it true that all algebraic values of the $j$-invariant have a real Galois conjugate?
This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it ...
- 13.6k
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Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality
The $j$-function (a.k.a. $j$-invariant up to a factor $1728$) is most often only considered for arguments where it yields real values.
It is well known that for $a,b,n\in\mathbb N$, $j\left(\dfrac {a+...
- 13.6k
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Taylor series of incomplete elliptic integral of first kind
Say I have an incomplete elliptic integral of first kind of the form
$$F(\varphi(z), k(z))=\int_0^{\varphi(z)} \frac{d \theta}{\sqrt{1-k(z)^2 \sin ^2 \theta}}$$
where each arguments are function of ...
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How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?
The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants")
are defined for $n>0$ by
$$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...
- 13.6k
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252
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Can an ellipse roll down a tilted sine curve without jumping?
Background
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.
In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...
- 5,065
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When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
...
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334
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A real-valued analogue of the Weierstrass $\wp$ Function
I am interested in the following function:
$$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$
This function is analogous to the Weierstrass $\wp$ function, the only ...
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How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?
Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows:
$$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$
$$\...
- 317
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Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?
Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals
$$\operatorname{sn}^2u,\operatorname{...
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258
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Can we integrate arbitrary rational functions of Jacobian elliptic functions?
We can integrate arbitrary rational functions of the trigonometric functions because of the tangent half-angle substitution (https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution). This led ...
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Asymptotic for Ramanujan's $\tau$-function
The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.
Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
- 317
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Decay estimates for simple elliptic equations
Let
and let $p(|z|)$ be the radial solution of the following equation
$$
\Delta p + 4q = 0\quad \text{in } \mathbb{R}^n
$$
where
$n\geq 2$,
$0<\alpha<1$,
$q \triangleq q(|z|) = \frac 1{1+ |z|^...
- 569
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Where does the Weierstrass expansion of $\operatorname{sn}$ come from?
In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$')
$$\operatorname{sn}u=\frac{B}{A}$$
where $...
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An identity for Weierstrass elliptic functions evaluation
Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE:
$$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$
and we assume $e_1=\frac{2-c}3>...
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Algebraic dependence of the elliptic functions
Let $\{f_i\}_{i=1}^{n}$ be $n$ elliptic functions in $\mathbb{C}$. We say that $f_1, \dots, f_n$ are algebraic dependent over $\mathbb{C}$ if there exists a polynomial of $n$ variables with constant ...
- 1,726
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Hecke operators acting on the Weierstrass $\wp$-function
Roughly speaking, my question is the following: Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function, where $\tau \in \mathbf{H}$ and $z \in \mathbf{C}$. If $p$ is a prime number, can we define $T_p\,\...
- 2,039
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Exact value of $0<\wp^{-1}(2^{-{2/3}})<2$ with $\wp$ the Weierstrass elliptic function
I am investigating solutions to the differential equation
$$\ddot{y}(t)=6y(t)^2,\dot{y}(0), y(0)=y_0>0.\tag{1}\label{1} $$
Let $\wp(t)$ be the Weierstrass elliptic function with elliptic invariants ...
- 946
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The origin and use of the term "equianharmonic" (elliptic function)
This question has been posted on History of Science and Mathematics stack exchange, but there was no answer or comments there.
In Weierstrass notation, the principal elliptic function $\wp$ is a ...
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What is the surface area of the finite part of the Cayley nodal cubic surface?
The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
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How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$
Let $\wp(u) = \frac{1}{u^2} + \sum\limits_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function.
My question is, how can I calculate $\wp(...
- 1,504
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1
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758
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Approximation of Incomplete elliptic integral of first kind
How can we represent F(x,m) in the infinte polynominal of x,m?
(Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia)
More ...
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Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?
Background
Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
- 5,065
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Explicit solution of the Lamé equation for n=1
The Jacobi form of Lamé equation is given by
\begin{equation}
\left(\frac{d^2 }{du^2} - n(n+1)k^2 \operatorname{sn}^{2}(u)-E\right)\Psi (u) = 0,
\end{equation}
where $k\in(0, 1)$ is parameter ...
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Closed form roots for polynomial $x^9 + ax^6 + bx^5 + cx^3 + d = 0$
I know about Abel–Ruffini theorem, but I have a polynomial of special form. From "Beyond the Quartic Equation" by R.B. King (a very interesting book, btw) I've learned about Tschirnhaus ...
- 365
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Elliptic functions
The Weierstrass $\wp$-function is given by
$$
\wp(z,\tau)=\frac{1}{z^{2}}+\sum_{(k,l)\neq (0,0)}\left(\frac{1}{(z-(k\tau+l))^{2}}-\frac{1}{(k\tau+l)^{2}}\right).
$$
Let $\lambda$ be primitive $n$th ...
- 417
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Resources on the stationary Schrödinger equation with the soliton potential
I am studying the following Lamé equation in the Jacobi form
\begin{equation}
-\frac{d^2 v}{dx^2} - \left(2k^2 \operatorname{cn}^{2}\left [x\;|\;k\right ]\right)v = \lambda v,
\end{equation}
...
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Can a doubly periodic function be locally univalent?
I am looking for a meromorphic doubly periodic function such that the function is locally univalent.
A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as
$$\wp(...
- 1,360
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1
answer
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How to work out this elliptic function?
Let $f(x) = \sum\limits_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$
If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange ...
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What is the analogue of the Jacobi theta function in the Weyl representation?
It is known (see for example the associated Wikipedia entry) that the Jacobi theta function
$$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$
arises from a certain ...
- 171
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0
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38
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Estimatives for elliptic systems involving the laplacian
Considering the problem
\begin{equation}
\left\{
\begin{array}[c]{11}
\Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\
\Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\
\end{...
- 101
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votes
1
answer
703
views
Generating function of the product of Legendre polynomials
The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
- 43
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2
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747
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Can the theory of elliptic functions developed from purely geometric considerations?
I always had this question, but was unable to get a definitive answer to it.
There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...
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474
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Regarding the Weierstrass $\wp$-function of the hexagonal lattice
Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$,
$$
\wp'(z)^2 = 4 \wp(z)^3 - 1,
$$
I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are
$$
\frac{\...
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0
answers
402
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How to determine the closed form of this Fourier series?
Consider the Series
$$
S(z) \equiv \sum_{n \in \mathbb{Z}, n \ne 0} \frac{ 1 }{ \sin n\pi \tau \sin 2n \pi \tau } e^{2\pi i n z} \ , \quad \operatorname{Im}\tau > 0
$$
I am trying to find its ...
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Infinite sum of iterated integrals of matrix products
Originally asked over at Stackexchange (https://math.stackexchange.com/questions/4169812/infinite-sum-of-iterated-integrals-of-matrix-products), but this forum was deemed more appropriate.
The problem:...
